# ARCHIMEDES (c.287 – c.212 BC)

Third Century BCE – Syracuse (a Greek city in Sicily)

‘Archimedes’ Screw – a device used to pump water out of ships and to irrigate fields’ Archimedes investigated the principles of static mechanics and pycnometry (the measurement of the volume or density of an object). He was responsible for the science of hydrostatics, the study of the displacement of bodies in water.

### Archimedes’ Principle

Buoyancy – ‘A body fully or partially immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body’
The upthrust (upward force) on a floating object such as a ship is the same as the weight of water it displaces. The volume of the displaced liquid is the same as the volume of the immersed object. This is why an object will float. When an object is immersed in water, its weight pulls it down, but the water, as Archimedes realised, pushes back up with a force that is equal to the weight of water the object pushes out-of-the-way. The object sinks until its weight is equal to the upthrust of the water, at which point it floats.
Objects that weigh less than the water displaced will float and objects that weigh more will sink. Archimedes showed this to be a precise and easily calculated mathematical principle.

Syracuse’s King Hiero, suspecting that the goldsmith had not made his crown of pure gold as instructed, asked Archimedes to find out the truth without damaging the crown.

Archimedes first immersed in water a piece of gold that weighed the same as the crown and pointed out the subsequent rise in water level. He then immersed the crown and showed that the water level was higher than before. This meant that the crown must have a greater volume than the gold, even though it was the same weight. Therefore it could not be pure gold and Archimedes thus concluded that the goldsmith had substituted some gold with a metal of lesser density such as silver. The fraudulent goldsmith was executed.

Archimedes came to understand and explain the principles behind the compound pulley, windlass, wedge and screw, as well as finding ways to determine the centre of gravity of objects.
He showed that the ratio of weights to one another on each end of a balance goes down in exact mathematical proportion to the distance from the pivot of the balance.

Perhaps the most important inventions to his peers were the devices created during the Roman siege of Syracuse in the second Punic war.

He was killed by a Roman soldier during the sack of the city. Pi

‘All circles are similar and the ratio of the circumference to the diameter of a circle is always the same number, known as the constant, Pi’ The Greek tradition disdained the practical.  Following PLATO the Greeks believed pure mathematics was the key to the perfect truth that lay behind the imperfect real world, so that anything that could not be completely worked out with a ruler and compass and elegant calculations was not true.

In the eighteenth century AD the Swiss mathematician LEONHARD EULER was the first person to use the letter  Π , the initial letter of the Greek word for perimeter, to represent this ratio.

The earliest reference to the ratio of the circumference of a circle to the diameter is an Egyptian papyrus written in 1650 BCE, but Archimedes first calculated the most accurate value.

He calculated Pi to be 22/7, a figure which was widely used for the next 1500 years. His value lies between 3 1/2 and 3 10/71, or between 3.142 and 3.141 accurate to two decimal places.

‘The Method of Exhaustion – an integral-like limiting process used to compute the area and volume of two-dimensional lamina and three-dimensional solids’

Archimedes realised how much could be achieved through practical approximations, or, as the Greeks called them, mechanics. He was able to calculate the approximate area of a circle by first working out the area of the biggest hexagon that would fit inside it and then the area of the smallest that would fit around it, with the idea in mind that the area of the circle must lie approximately halfway between.

By going from hexagons to polygons with 96 sides, he could narrow the margin for error considerably. In the same way he worked out the approximate area contained by all kinds of different curves from the area of rectangles fitted into the curve. The smaller and more numerous the rectangles, the closer to the right figure the approximation became.

This is the basis of what thousands of years later came to be called integral calculus.
Archimedes’ reckonings were later used by Kepler, Fermat, Leibniz and Newton.

In his treatise ‘On the Sphere and the Cylinder’, Archimedes was the first to deduce that the volume of a sphere is 4/3 Pi r3  where r  is the radius.

He also deduced that a sphere’s surface area can be worked out by multiplying that of its greatest circle by four; or, similarly, a sphere’s volume is two-thirds that of its circumscribing cylinder.

Like the square and cube roots of 2, Pi is an irrational number; it takes a never-ending string of digits to express Pi as a number.
It is impossible to find the exact value of Pi – however, the value can be calculated to any required degree of accuracy.
In 2002 Yasumasa Kanada (b.1949) of Tokyo University used a supercomputer with a memory of 1024GB to compute the value to 124,100,000,000 decimal places. It took 602 hours to perform the calculation.

(image source) TIMELINE

###### Related sites
• Pi (math.com)

# GEORG SIMON OHM (1789-1854)

1827 – Germany

‘The electric current in a conductor is proportional to the potential difference’

In equation form, V = IR, where V is the potential difference, I is the current and R is a constant called resistance. Ohm’s law links voltage (potential difference) with current and resistance and the scientists VOLTA, AMPERE and OHM.

Ohm is now honoured by having the unit of electrical resistance named after him.
If we use units of VI and R, Ohm’s law can be written in units as:

volts = ampere × ohm  TIMELINE ELECTRICITY

# ALBERT EINSTEIN (1879-1955)

1905 – Switzerland

1. ‘the relativity principle: All laws of science are the same in all frames of reference.
2. constancy of the speed of light: The speed of light in a vacuüm is constant and is independent of the speed of the observer’

The laws of physics are identical to different spectators, regardless of their position, as long as they are moving at a constant speed in relation to each other. Above all the speed of light is constant. Classical laws of mechanics seem to be obeyed in our normal lives because the speeds involved are insignificant.

Newton’s recipe for measuring the speed of a body moving through space involved simply timing it as it passed between two fixed points. This is based on the assumptions that time is flowing at the same rate for everyone – that there is such a thing as ‘absolute’ time, and that two observers would always agree on the distance between any two points in space.
The implications of this principle if the observers are moving at different speeds are bizarre and normal indicators of velocity such as distance and time become warped. Absolute space and time do not exist. The faster an object is moving the slower time moves. Objects appear to become shorter in the direction of travel. Mass increases as the speed of an object increases. Ultimately nothing may move faster than or equal to the speed of light because at that point it would have infinite mass, no length and time would stand still.

‘The energy (E) of a body equals its mass (m) times the speed of light (c) squared’

This equation shows that mass and energy are mutually convertible under certain conditions.

The mass-energy equation is a consequence of Einstein’s theory of special relativity and declares that only a small amount of atomic mass could unleash huge amounts of energy.

Two of his early papers described Brownian motion and the ‘photoelectric’ effect (employing PLANCK’s quantum theory and helping to confirm Planck’s ideas in the process).

1915 – Germany

‘Objects do not attract each other by exerting pull, but the presence of matter in space causes space to curve in such a manner that a gravitational field is set up. Gravity is the property of space itself’

From 1907 to 1915 Einstein developed his special theory into a general theory that included equating accelerating forces and gravitational forces. This implies light rays would be bent by gravitational attraction and electromagnetic radiation wavelengths would be increased under gravity. Moreover, mass and the resultant gravity, warps space and time, which would otherwise be ‘flat’, into curved paths that other masses (e.g. the moons of planets) caught within the field of the distortion follow. The predictions from special and general relativity were gradually proven by experimental evidence.

Einstein spent much of the rest of his life trying to devise a unified theory of electromagnetic, gravitational and nuclear fields.   TIMELINE QUANTUM MECHANICS

# ALBERT MICHELSON (1852-1931) EDWARD MORLEY (1838-1923)

1887 – USA

‘The aim of the experiment was to measure the effect of the Earth’s motion on the speed of light’

This celebrated experiment found no evidence of there being an effect.    TIMELINE RELATIVITY

# WERNER HEISENBERG (1901- 76)

1927 – Germany

‘It is impossible to determine exactly both the position and momentum of a particle (such as an electron) simultaneously’

The principle excludes the existence of a particle that is stationary.

To measure both the position and momentum ( momentum = mass × velocity ) of a particle simultaneously requires two measurements: the act of performing the first measurement will disturb a particle and so create uncertainty in the second measurement.
Thus the more accurately a position is known; the less accurately can the momentum be determined.

The disturbance is so small it can be ignored in the macroscopic world, but is quite dramatic for particles in the microscopic world.
MAX BORN’S ‘probabilistic’ interpretation, expressed at about the same time, concerning the likelihood of finding a particle at any point through probability defined by the amplitude of its associated wave, led to similar conclusions.

The uncertainty principle also applies to energy and time. A particle’s kinetic energy cannot be measured with complete precision either.

Heisenberg suggested the model of the proton and neutron being held together in the nucleus of the atom after the work of JAMES CHADWICK who discovered the neutron in 1932.

Heisenberg decided to try to develop a new model of the atom, more fundamentally based on quantum theory that worked for all atoms. He believed the approach of trying to visualise a physical model of the atom was destined to fail because of the paradoxical wave-particle nature of electrons.

Every particle has an associated wave. The position of a particle can be precisely located where the wave’s undulations are most intense. But where the wave’s undulations are most intense, the wavelength is also at its most ill-defined, and the velocity of the associated particle is impossible to determine. Similarly, a particle with a well-defined wavelength has a precise velocity but a very ill-defined position.

Since the orbits of electrons could not be observed, he decided to ignore them and focus instead on what could be observed and measured; namely the energy they emitted and absorbed, as shown in the spectral lines. He tried to devise a mathematical way of representing the orbits of electrons, and to use this as a way of predicting the atomic features shown up in the spectral lines.
He showed that matrix mechanics could account for many of the properties of atoms, including those with more than one electron.

Together with PAUL DIRAC, Pascual Jordan created a new set of equations based on the rival theories of Schrödinger and Heisenberg, which they called ‘transformation theory’. Whilst studying these equations, Heisenberg noticed the paradox that measurements of position and velocity (speed and direction) of particles taken at the same time gave imprecise results. He believed that this uncertainty was a part of the nature of the sub-atomic world. The act of measuring the velocity of a subatomic particle will change it, making the simultaneous measurement of its position invalid.

An unobserved object is both a particle and a wave. If an experimenter chooses to measure the object’s velocity, the object will transform itself into a wave. If an experimenter chooses to measure its position, it will become a particle. By choosing to observe either one thing or the other, the observer is actually affecting the form the object takes.
The practical implication of this is that one can never predict where an electron will be at a precise moment, one can only predict the probability of its being there.   NEXT QUANTUM ELECTRODYNAMICS

# PAUL DIRAC (1902- 84)

1928 – UK

‘Every fundamental particle has an antiparticle – a mirror twin with the same mass but opposite charge’

‘It appears that the simplest Hamiltonian for a point-charge electron satisfying the requirements of both relativity and the general transformation theory leads to an explanation of all duplexity phenomena without further assumption’

1931 – UK

‘A magnetic monopole is analogous to electric charge’

A magnetic monopole is a hypothetical particle that carries a basic magnetic charge – in effect, a single north or south magnetic pole acting as a free particle.

Until recently no one has observed a monopole.   TIMELINE QUANTUM ELECTRODYNAMICS

# GEORGE FITZGERALD (1851-1901) HENDRICK LORENTZ (1853-1928)

1890 – Ireland
1904 – Holland

‘A moving object appears to contract’

The contraction is negligible unless the object’s speed is close to the speed of light.

In 1890 Fitzgerald suggested that an object moving through space would shrink slightly in its direction of travel by an amount dependent on its speed.

In 1904 Lorentz independently studied this problem from an atomic point of view and derived a set of equations to explain it. A year later, Einstein derived Lorentz’s equations independently from his special theory of relativity.  TIMELINE RELATIVITY